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Izv. Akad. Nauk SSSR Ser. Mat., 1968, Volume 32, Issue 5, Pages 1138–1146 (Mi izv2509)  

This article is cited in 23 scientific papers (total in 23 papers)

Some general questions in the theory of the Riemann boundary problem

I. B. Simonenko


Abstract: In this paper we investigate the Riemann boundary problem
$$ \Phi^+(t)=G(t)\Phi^-(t)+g(t) $$
for $n$ pairs of functions. The solutions $\Phi^\pm$ are to belong to the classes $E_p^\pm$; the given function g belongs to the class $L_p$ $(1<p<\infty)$. We enlarge the class of coefficients $G$ for which the Noether theory remains valid. In the case $n=1$, $p=2$, necessary and sufficient conditions for Noetherianness are obtained. It is shown that the class of matrix-functions which admit factorization coincides with the class for which the Noether theory is valid. In the case $n=1$ it is shown that one of the defect numbers is zero.

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English version:
Mathematics of the USSR-Izvestiya, 1968, 2:5, 1091–1099

Bibliographic databases:

UDC: 517.9
MSC: 30F20, 30E25, 28B20, 46E30, 47A56, 26B35
Received: 03.01.1968

Citation: I. B. Simonenko, “Some general questions in the theory of the Riemann boundary problem”, Izv. Akad. Nauk SSSR Ser. Mat., 32:5 (1968), 1138–1146; Math. USSR-Izv., 2:5 (1968), 1091–1099

Citation in format AMSBIB
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\by I.~B.~Simonenko
\paper Some general questions in the theory of the Riemann boundary problem
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1968
\vol 32
\issue 5
\pages 1138--1146
\mathnet{http://mi.mathnet.ru/izv2509}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=235135}
\zmath{https://zbmath.org/?q=an:0165.16703|0186.13601}
\transl
\jour Math. USSR-Izv.
\yr 1968
\vol 2
\issue 5
\pages 1091--1099
\crossref{https://doi.org/10.1070/IM1968v002n05ABEH000706}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. I. M. Spitkovsky, “Some estimates for the partial indices of measurable matrix-valued functions”, Math. USSR-Sb., 39:2 (1981), 207–226  mathnet  crossref  mathscinet  zmath  isi
    2. G. S. Litvinchuk, I. M. Spitkovsky, “Sharp estimates of defect numbers of a generalized Riemann boundary value problem, factorization of hermitian matrix-valued functions and some problems of approximation by meromorphic functions”, Math. USSR-Sb., 45:2 (1983), 205–224  mathnet  crossref  mathscinet  zmath
    3. Yu. I. Karlovich, V. G. Kravchenko, “An algebra of singular integral operators with piecewise-continuous coefficients and a piecewise-smooth shift on a composite contour”, Math. USSR-Izv., 23:2 (1984), 307–352  mathnet  crossref  mathscinet  zmath
    4. S. M. Grudskii, “Singular integral equations and the Riemann boundary value problem with infinite index in the space $L_p(\Gamma,\omega)$”, Math. USSR-Izv., 26:1 (1986), 53–76  mathnet  crossref  mathscinet  zmath
    5. N. L. Vasilevskii, “On an algebra connected with Toeplitz operators in radial tube domains”, Math. USSR-Izv., 30:1 (1988), 71–88  mathnet  crossref  mathscinet  zmath
    6. I. M. Spitkovsky, “On a vectorial Riemann boundary value problem with infinite defect numbers, and related factorization of matrix-valued functions”, Math. USSR-Sb., 63:2 (1989), 521–538  mathnet  crossref  mathscinet  zmath
    7. Yu. I. Karlovich, I. M. Spitkovsky, “Factorization of almost periodic matrix-valued functions and the Noether theory for certain classes of equations of convolution type”, Math. USSR-Izv., 34:2 (1990), 281–316  mathnet  crossref  mathscinet  zmath
    8. Albrecht Böttcher, Yuri I. Karlovich, “Toeplitz and singular integral operators on Carleson curves with logarithmic whirl points”, Integr equ oper theory, 22:2 (1995), 127  crossref  mathscinet  zmath  isi
    9. I. Feldman, I. Gohberg, N. Krupnik, “On explicit factorization and applications”, Integr equ oper theory, 21:4 (1995), 430  crossref  mathscinet  zmath  isi
    10. L. P. Castro, F.-O. Speck, “On the Characterization of the Intermediate Space in Generalized Factorizations”, Math Nachr, 176:1 (1995), 39  crossref  mathscinet  zmath  isi
    11. A. Böttcher, Yu. I. Karlovich, V. S. Rabinovich, “Emergence, persistence, and disappearance of logarithmic spirals in the spectra of singular integral operators”, Integr equ oper theory, 25:4 (1996), 406  crossref  mathscinet  isi  elib
    12. Albrecht Böttcher, Yuri I. Karlovich, “Submultpilicative fuctions and spectral theory of toeplitz operators”, Integral Transforms and Special Functions, 4:1-2 (1996), 181  crossref
    13. A. Yu. Karlovich, “The index of singular integral operators in reflexive Orlicz spaces”, Math. Notes, 64:3 (1998), 330–341  mathnet  crossref  crossref  mathscinet  zmath  isi
    14. M.A. Bastos, Yu.I. Karlovich, A.F. dos Santos, P.M. Tishin, “The Corona Theorem and the Existence of Canonical Factorization of Triangular AP-Matrix Functions”, Journal of Mathematical Analysis and Applications, 223:2 (1998), 494  crossref
    15. J. A. Ball, Yu. I. Karlovich, L. Rodman, I. M. Spitkovsky, “Sarason interpolation and Toeplitz corona theorem for almost periodic matrix functions”, Integr equ oper theory, 32:3 (1998), 243  crossref
    16. C. J. Bishop, A. Böttcher, Yu. I. Karlovich, I. Spitkovsky, “Local Spectra and Index of Singular Integral Operators with Piecewise Continuous Coefficients on Composed Curves”, Math Nachr, 206:1 (1999), 5  crossref  elib
    17. Torsten Ehrhardt, Frank-Olme Speck, “Transformation techniques towards the factorization of non-rational 2×2 matrix functions”, Linear Algebra and its Applications, 353:1-3 (2002), 53  crossref
    18. M.A. Bastos, Yu.I. Karlovich, A.F. dos Santos, “Oscillatory Riemann–Hilbert problems and the corona theorem”, Journal of Functional Analysis, 197:2 (2003), 347  crossref
    19. V. V. Simonyan, “On connection of one class of one-dimensional pseudodifferential operators with singular integral operators”, Uch. zapiski EGU, ser. Fizika i Matematika, 2009, no. 2, 8–15  mathnet
    20. Karlovich A.Yu., “Singular Integral Operators on Variable Lebesgue Spaces with Radial Oscillating Weights”, Operator Algebras, Operator Theory and Applications, Operator Theory Advances and Applications, 195, 2010, 185–212  isi
    21. Karlovich A.Yu., “Singular Integral Operators on Variable Lebesgue Spaces over Arbitrary Carleson Curves”, Topics in Operator Theory: Operators, Matrices and Analytic Functions, Operator Theory Advances and Applications, 1, 2010, 321–336  isi
    22. G. Mishuris, S. Rogosin, “An asymptotic method of factorization of a class of matrix functions”, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 470:2166 (2014), 20140109  crossref
    23. A. G. Kamalian, I. M. Spitkovsky, “On the Fredholm Property of a Class of Convolution-Type Operators”, Math. Notes, 104:3 (2018), 404–416  mathnet  crossref  crossref  isi  elib
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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